H\"ormander's Inequality and Point Evaluations in de Branges Space

TL;DR

This paper extends Hörmander's inequality to de Branges spaces, analyzing decay rates of entire functions and studying point evaluation functionals, thereby generalizing recent results in the theory of model spaces.

Contribution

It generalizes Hörmander's inequality to de Branges spaces and investigates point evaluation functionals within this framework.

Findings

Extended Hörmander's inequality to de Branges spaces.

Analyzed decay rates of functions in de Branges spaces.

Studied extremal functions and point evaluations, generalizing recent results.

Abstract

Let $f$ be an entire function of finite exponential type less than or equal to $\sigma$ which is bounded by $1$ on the real axis and satisfies $f(0) = 1$. Under these assumptions H\"ormander showed that $f$ cannot decay faster than $\cos(\sigma x)$ on the interval $(-\pi/\sigma,\pi/\sigma)$. We extend this result to the setting of de Branges spaces with cosine replaced by the real part of the associated Hermite-Biehler function. We apply this result to study the point evaluation functional and associated extremal functions in de Branges spaces (equivalently in model spaces generated by meromorphic inner functions) generalizing some recent results of Brevig, Chirre, Ortega-Cerd\`a, and Seip.